Standard deviation | CAIE 9709 Probability & Statistics 1 Notes

Probability & Statistics 1

Mathematics9709

Variance first, then standard deviation

Key idea

Variance is the average squared distance from the mean. Standard deviation is its square root, so it returns to the original unit.

\operatorname{Var}(x)=\frac{\sum x^2}{n}-\bar x^2

Symbols

$n$ = number of values (observations)
$\sum x$ = sum of the values (data unit)
$\sum x^2$ = sum after squaring each value (data unit squared)
$\bar x$ = mean (data unit)

The variance is in squared units. Take its positive square root to return to the data unit.

\operatorname{SD}(x)=\sqrt{\operatorname{Var}(x)}

Standard deviation can never be negative.
It is zero only when every value is the same.
It uses every value, so an extreme value can change it strongly.

Worked example

Standard deviation of four values

For 2, 4, 4 and 6, $n=4$ , $\sum x=16$ and $\sum x^2=72$ .

The mean is 4. Substituting before taking the square root gives the variance.

\operatorname{Var}=\frac{72}{4}-4^2=2

Answer

\operatorname{SD}=\sqrt2=1.41\ \text{(3 s.f.)}

Common mistake

\sum x^2

means square each value, then add. It is not

(\sum x)^2

. Also check whether the question asks for variance or standard deviation; only SD needs the final square root.

Frequency tables use weighted totals

\bar x=\frac{\sum fx}{\sum f}

\operatorname{Var}=\frac{\sum fx^2}{\sum f}-\bar x^2

Build two weighted columns. Calculate $fx$ and $fx^2$ . The second means $f\times x^2$ .
Add the columns. Record $\sum f$ , $\sum fx$ and $\sum fx^2$ clearly.
Keep the exact mean. Do not round $\bar x$ before squaring it.
Root only at the end. The formula inside the root is the variance.

Worked example

A discrete frequency table

Values 5, 15 and 25 have frequencies 2, 5 and 3. Then $\sum f=10$ , $\sum fx=160$ and $\sum fx^2=3050$ .

\operatorname{Var}=\frac{3050}{10}-16^2=49

Answer

\operatorname{SD}=7

For grouped continuous data, use class midpoints for $x$ . The result is an estimate because the exact values inside each class are unknown.

Use the calculator as the fast main route

For a straightforward raw-data, frequency-table or grouped-data question on a calculator-permitted paper, one-variable statistics is normally the fastest main method. You do not need to calculate every total by hand first.

Enter values in the one-variable statistics mode.
Enter each value once; when frequencies are given, use the frequency column. For grouped data, enter class midpoints as the values and class frequencies beside them.
Read the population standard deviation $\sigma_x$ , not the sample value $s_x$ .
Write the requested mean, variance or SD from the display, rounded only at the end. For variance, square $\sigma_x$ .

Use the calculator as the main route when the full data or a frequency table is available. Switch to equations and totals when the question gives totals or coded totals, asks you to show a result, combines data sets, asks for a missing value, or needs visible method marks.

Use the range for a quick full-width description.
Use the IQR when extremes should have less effect.
Use standard deviation when every value should contribute and the mean is a suitable centre.

Your turnIndependent check [5 marks]

Values 1, 3, 5 and 7 have frequencies 2, 1, 4 and 3. Find the mean, variance and population standard deviation.

Show worked answer

$\sum f=10$ , $\sum fx=46$ and $\sum fx^2=258$ , so $\bar x=4.6$ .

Answer

25.8-4.6^2=4.64

\sqrt{4.64}=2.154\ldots

\operatorname{SD}\approx2.15

Variance first, then standard deviation

Key idea

Variance is the average squared distance from the mean. Standard deviation is its square root, so it returns to the original unit.

\operatorname{Var}(x)=\frac{\sum x^2}{n}-\bar x^2

Symbols

$n$ = number of values (observations)
$\sum x$ = sum of the values (data unit)
$\sum x^2$ = sum after squaring each value (data unit squared)
$\bar x$ = mean (data unit)

The variance is in squared units. Take its positive square root to return to the data unit.

\operatorname{SD}(x)=\sqrt{\operatorname{Var}(x)}

Standard deviation can never be negative.
It is zero only when every value is the same.
It uses every value, so an extreme value can change it strongly.

Worked example

Standard deviation of four values

For 2, 4, 4 and 6, $n=4$ , $\sum x=16$ and $\sum x^2=72$ .

The mean is 4. Substituting before taking the square root gives the variance.

\operatorname{Var}=\frac{72}{4}-4^2=2

Answer

\operatorname{SD}=\sqrt2=1.41\ \text{(3 s.f.)}

Common mistake

\sum x^2

means square each value, then add. It is not

(\sum x)^2

. Also check whether the question asks for variance or standard deviation; only SD needs the final square root.

Frequency tables use weighted totals

\bar x=\frac{\sum fx}{\sum f}

\operatorname{Var}=\frac{\sum fx^2}{\sum f}-\bar x^2

Build two weighted columns. Calculate $fx$ and $fx^2$ . The second means $f\times x^2$ .
Add the columns. Record $\sum f$ , $\sum fx$ and $\sum fx^2$ clearly.
Keep the exact mean. Do not round $\bar x$ before squaring it.
Root only at the end. The formula inside the root is the variance.

Worked example

A discrete frequency table

Values 5, 15 and 25 have frequencies 2, 5 and 3. Then $\sum f=10$ , $\sum fx=160$ and $\sum fx^2=3050$ .

\operatorname{Var}=\frac{3050}{10}-16^2=49

Answer

\operatorname{SD}=7

For grouped continuous data, use class midpoints for $x$ . The result is an estimate because the exact values inside each class are unknown.

Use the calculator as the fast main route

Enter values in the one-variable statistics mode.
Enter each value once; when frequencies are given, use the frequency column. For grouped data, enter class midpoints as the values and class frequencies beside them.
Read the population standard deviation $\sigma_x$ , not the sample value $s_x$ .
Write the requested mean, variance or SD from the display, rounded only at the end. For variance, square $\sigma_x$ .

Use the range for a quick full-width description.
Use the IQR when extremes should have less effect.
Use standard deviation when every value should contribute and the mean is a suitable centre.

Your turnIndependent check [5 marks]

Values 1, 3, 5 and 7 have frequencies 2, 1, 4 and 3. Find the mean, variance and population standard deviation.

Show worked answer

$\sum f=10$ , $\sum fx=46$ and $\sum fx^2=258$ , so $\bar x=4.6$ .

Answer

25.8-4.6^2=4.64

\sqrt{4.64}=2.154\ldots

\operatorname{SD}\approx2.15